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In this paper, we consider the dynamics of a prey–predator model with logistic growth incorporating refuge in the prey and cooperation among predators population. Furthermore, multiplicative Allee effect in the prey growth is added to account from biological and mathematical perspectives. First, the existence and stability of equilibria of the model are discussed. Next, the existence of several kinds of bifurcation is provided and also studied the direction and stability of Hopf bifurcation. In addition, we study the influence of hunting cooperation on the model analytically and numerically, and find that the hunting cooperation cannot only reduce the density of prey population, but also destabilize the system dynamics irrespective of Allee effect. We choose also the impact of refuge on the model numerically, and explore that refuge stabilize the system dynamics. Moreover, the comparison between the dynamics of strong and weak Allee effect is taken into consideration.

One of the most important factors influencing animal growth is non-genetics, which includes factors like nutrition, management and environmental conditions. By consuming their prey, predators can directly affect their ecology and evolution, but they can also have an indirect impact by affecting their prey’s nutrition and reproduction. Preys used to change their habitats, their foraging and vigilance habits as anti-predator responses. Cooperation during hunting by the predators develops significant fear in their prey which indirectly affects their nutrition. In this work, we propose a two-species stage-structured predator–prey system where the prey are classified into juvenile and mature prey. We assume that the conversion of juvenile prey to matured prey is affected by the fear of predation risk. Non-negativity and boundedness of the solutions are demonstrated theoretically. All the biologically feasible equilibrium states are determined, and their stabilities are analyzed. The role of various important factors, e.g. hunting cooperation rate, predation rate and rate of fear, on the system dynamics is discussed. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MatCont 7.3. Finally, the proposed model is extended into a harvesting model under quadratic harvesting strategy and the associated control problem has been analyzed for optimal harvesting.

Predator foraging facilitation or cooperative hunting increases per predator consumption rate as predator density increases. This affects predator extinction in a prey–predator interaction model when the predator density is low. This is an indication of Allee effect in predator’s growth rate. Here, we take a Gause type model with a generalized type II functional response which depends on both prey and predator densities. We also assume that prey’s growth is subjected to Allee effect. Strong Allee effect in prey’s growth rate enhances the stability of the coexisting steady state. A region is found in a two-parameter plane where the coexisting steady state is a global attractor when the prey’s growth is subjected to weak Allee effect. In addition, codimension two bifurcation points (cusp and Bogdanov–Takens points) have also been found in the bifurcation diagram.

In this paper, we are concerned with a diffusive predator–prey model where the functional response follows the predator cooperation in hunting and the growth of the prey obeys the Allee effect. Firstly, the existence and stability of the positive equilibrium are explicitly determined by the local system parameters. It is shown that the ability of the hunting cooperation can affect the existence of the positive equilibrium and stability, and the intrinsic growth rate of the predator population does not affect the existence of the positive equilibrium, but affects the stability. Then the diffusion-driven Turing instability is investigated and the Turing bifurcation value is obtained, and we conclude that when the ability of the cooperation in hunting is weaker than some critical value, there is no Turing instability. The standard weakly nonlinear analysis method is employed to derive the amplitude equations of the Turing bifurcation, which is used to predict the types of the spatial patterns. And it is found that in the Turing instability region, with the parameter changing from approaching Turing bifurcation value to approaching Hopf bifurcation value, spatial patterns emerge from spot, spot-stripe to stripe. Finally, the numerical simulations are used to support the analytical results.

With both hunting cooperation and Allee effects in predators, a predator–prey system was modeled as a planar cubic differential system with three parameters. The known work numerically plots the horizontal isocline and the vertical one with appropriately chosen parameter values to show the cases of two, one and no coexisting equilibria. Transitions among those cases with the rise of limit cycle and homoclinic loop were exhibited by numerical simulations. Although it is hard to obtain the explicit expression of coordinates, in this paper, we give the distribution of equilibria qualitatively, discuss all cases of coexisting equilibria, and obtain the Bogdanov–Takens bifurcation diagram to show analytical parameter conditions for those transitions. Our results give analytical conditions for not only the observed saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation but also the transcritical and pitchfork bifurcations at the predator-extinction equilibrium, which were not considered in the known work. Our analytic conditions provide a quantitative instruction to reduce the risk of predator extinction and promote the ecosystem diversity.

Understanding the effects of cooperative hunting among the predators is an important topic in ecology. We explore a prey–predator model with cooperative hunting among predators modeled by a Holling type-II functional response having saturated encounter rate. Bifurcation analysis is carried out on the temporal model to detect local bifurcations such as transcritical, saddle-node, Hopf bifurcation and global homoclinic bifurcation. A comprehensive numerical bifurcation analysis is performed on the Turing solutions of the corresponding diffusive model. Stable and unstable branches of mode specific Turing solutions and their dependence on parameters are identified. For suitable values of parameters, spatio-temporal chaos via period doubling cascade is observed when a given parameter traverses the Turing–Hopf region.

In prey–predator interaction, many factors, such as the fear effect, Allee effect, cooperative hunting, and group behavior, can influence the population dynamics. Hence, studying these factors in prey–predator makes the model more realistic. In this paper, we have proposed the prey–predator model having herd and Allee effect in prey population, where predators follow hunting cooperation. We have employed temporal analysis to examine the role of the Allee effect and hunting cooperation. Furthermore, we have extended the analysis to spatiotemporal analysis to examine the role of dispersal and the type of spatial structure formed by the population due to random movement. We first discuss the proposed model’s existence and positivity, then the stability of the existing equilibrium points through Routh–Hurwitz criteria. The temporal analysis is carried out through Hopf-bifurcation at the coexistence equilibrium point by considering the Allee threshold ($\alpha$), hunting cooperation ($\gamma$), and attack rate ($\beta$) as controlled parameters. With the addition of diffusion to the model, we examine the spatial model stability and derive the Turing instability condition, which will give rise to various Turing patterns. Finally, numerical simulations are performed to validate the analytical results. The theoretical study and numerical simulation results demonstrate that the Allee effect, hunting cooperation, and diffusion coefficient are sensitive parameters to the model’s stability.

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results.

In this paper, we scrutinize the dynamics of a temporal and spatiotemporal prey–predator model incorporating the fear effect on prey and team hunting by the predator. Additionally, we explore the influence of delayed anti-predation response. The analysis includes discussions on well-posedness, local stability, and various bifurcations such as saddle-node, transcritical, Hopf and Bogdanov–Takens bifurcations. The impact of fear cost and delay parameters on model dynamics is investigated by considering them as bifurcation parameters. We investigate how bifurcation values change with varying parameters by exploring different bi-parameter planes. It is observed that the system transitions into chaotic behavior through Hopf bifurcation for significant anti-predation response delay. The positivity of the maximal Lyapunov exponent indicates the confirmed characteristics of chaotic behavior. Furthermore, within the spatiotemporal model framework, a thorough analysis of local and global stability is provided, including the establishment of criteria for identifying Turing instability in cases of self- and cross-diffusion. Various stationary and dynamic patterns are elucidated as diffusion coefficients vary, showcasing the diverse dynamics of the spatiotemporal model. In order to illustrate the dynamic characteristics of the system, a series of comprehensive numerical simulations are conducted. The discoveries outlined in this paper could prove advantageous for understanding the biological implications resulting from the examination of predator–prey relationships.

In this paper, we have investigated a spatial predator–prey model with hunting cooperation in predators. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of controlling parameters. Using extensive numerical simulations, we obtain complex patterns, namely, spotted pattern, stripe pattern and mixed pattern in the Turing domain, by varying the hunting cooperation parameter in predators and carrying capacity of preys. The results focus on the effect of hunting cooperation in pattern dynamics of a diffusive predator–prey model and help us in better understanding of the dynamics of the predator–prey interaction in real environments.

Fear of predation may assert privilege to prey species by restricting their exposure to potential predators, meanwhile it can also impose costs by constraining the exploration of optimal resources. A predator–prey model with the effect of fear, refuge, and hunting cooperation has been investigated in this paper. The system’s equilibria are obtained and their local stability behavior is discussed. The existence of Hopf-bifurcation is analytically shown by taking refuge as a bifurcation parameter. There are many ecological factors which are not instantaneous processes, and so, to make the system more realistic, we incorporate three discrete time delays: in the effect of fear, refuge and hunting cooperation, and analyze the delayed system for stability and bifurcation. Moreover, for environmental fluctuations, we further modify the delayed system by incorporating seasonality in the fear, refuge and cooperation. We have analyzed the seasonally forced delayed system for the existence of a positive periodic solution. In the support of analytical results, some numerical simulations are carried out. Sensitivity analysis is used to identify parameters having crucial impacts on the ecological balance of predator–prey interactions. We find that the rate of predation, fear, and hunting cooperation destabilizes the system, whereas prey refuge stabilizes the system. Time delay in the cooperation behavior generates irregular oscillations whereas delay in refuge stabilizes an otherwise unstable system. Seasonal variations in the level of fear and refuge generate higher periodic solutions and bursting patterns, respectively, which can be replaced by simple 1-periodic solution if the cooperation and fear are also allowed to vary with time in the former and latter situations. Higher periodicity and bursting patterns are also observed due to synergistic effects of delay and seasonality. Our results indicate that the combined effects of fear, refuge and hunting cooperation play a major role in maintaining a healthy ecological environment.

Predator foraging facilitation or hunting cooperation and the antipredator behavior of prey are essential mechanisms in evolutionary biology and ecology and may strongly influence the predator–prey dynamics. In a real-world scenario, this behavioral tendency is well documented, but less is known about how it could affect the dynamics between predator and prey. Here, we investigate the impact of the fear of predator on prey and the hunting cooperation in predator on the predator–prey dynamics, where the predator is assumed to be of generalist type. We observe that without fear, even with the high level of hunting cooperation, both populations may coexist, though the increasing level of hunting cooperation reduces the prey density at coexistence equilibrium. Moreover, increasing level of fear also destabilizes the system with and without hunting cooperation. Further, in the presence of hunting cooperation and fear effect, the model shows three different types of bistability phenomena: bistability between two coexisting equilibria, bistability between coexisting equilibria and prey-free equilibrium, and bistability between stable limit cycle and coexisting equilibria. In addition, saddle-node, Hopf, transcritical bifurcation of codimension one, Bautin (generalized Hopf), Bogdanov–Takens, and cusp bifurcation of codimension two are observed.

Depending on behavioral differences, reproductive capability and dependency, the life span of a species is divided mainly into two classes, namely immature and mature. In this paper, we have studied the dynamics of a predator–prey system considering stage structure in prey and the effect of predator-induced fear with two discrete time delays: maturation delay and fear response delay. We consider that predators cooperate during hunting of mature prey and also include its impact in fear term. The conditions for existence of different equilibria, their stability analysis are carried out for non-delayed system and bifurcation results are presented extensively. It is observed that the fear parameter has stabilizing effect whereas the cooperative hunting factor having destabilizing effect on the system via occurrence of supercritical Hopf-bifurcation. Further, we observe that the system exhibits backward bifurcation between interior equilibrium and predator free equilibrium and hence the situation of bi-stability occurs in the system. Thereafter, we differentiate the region of stability and instability in bi-parametric space. We have also studied the system’s dynamics with respect to maturation and fear response delay and observed that they also play a vital role in the system stability and occurrence of Hopf-bifurcation is shown with respect to both time delays. The system shows stability switching phenomenon and even higher values of fear response delay leads the system to enter in chaotic regime. The role of fear factor in switching phenomenon is discussed. Comprehensive numerical simulation and graphical presentation are carried out using MATLAB and MATCONT.

In this paper, we consider a modified Lasslie–Gower-type predator–prey model with the effect of hunting cooperation and favorable additional food for predator. We establish the conditions of positivity, boundedness, and permanence of solutions of the proposed model. Along with the trivial, predator free, prey free equilibrium points the system contains at most two coexistence equilibrium points. The system experiences the transcritical, saddle-node, Hopf, cusp, Bautin, and Bogdanov–Takens bifurcation depending on the model parameters. All the theoretical analyses are verified using numerical simulations. It is numerically established that the cooperation and extra food have high impact on the model dynamics.

In this paper, we examine the role of fear and hunting cooperation in a generalist predator–prey system in an imprecise environment. We investigated the positivity and boundedness of the solutions and determined the existence of equilibria and their local stability for the system. We analyzed the transcritical and Hopf bifurcation considering the uncertain parameter as a bifurcation parameter. Based on simulation results, both populations can survive at low to moderate values of imprecise parameters, but the prey population cannot survive at higher values of the imprecise parameter. Additionally, we observe fear, hunting cooperation, and generalist predators with an imprecise concept that illustrates the complex dynamics of the system.

The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity. Hunting cooperation in predators is one of the most remarkable features of the predation process, which benefits the predators by developing fear upon their prey. This study investigates the dynamical behavior of a modified LV-type predator–prey system with Michaelis–Menten-type harvesting of predators where predators adopt cooperation strategy during hunting. The ecologically feasible steady states of the system and their asymptotic stabilities are explored. The local codimension one bifurcations, viz. transcritical, saddle-node and Hopf bifurcations, that emerge in the system are investigated. Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation. A backward Hopf-bifurcation is detected when the harvesting effort is increased, which destabilizes the system by generating periodic solutions. The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient. Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations. Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased. We have also explored codimension two bifurcations viz. the generalized Hopf and the Bogdanov–Takens bifurcation exhibit by the system. To visualize the dynamical behavior of the system, numerical simulations are conducted using an ecologically plausible parameter set. The existence of the bionomic equilibrium of the model is analyzed. Moreover, an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.

Self-diffusion prerequisite is obtained as the spreading approach of biological populations. Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey–predator system. On the other side, the Allee effect among prey may cause the system to become unstable. In this paper, a diffusive prey–predator system with cooperative hunting and the weak Allee effect in prey populations is discussed. The linear stability and Hopf-bifurcation analysis had been used to examine the system’s stability. From the spatial stability of the system, the conditions for Turing instability have been derived. The multiple-scale analysis has been used to derive the amplitude equations of the system. The stability analysis of these amplitude equations leads to the formation of Turing patterns. Finally, numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D. The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impact on species distribution.